Realization Of (?)(Sp_q(N)) Arising From U_q(sp(N)) Via Jantzen Approaching

Jantzen gave the defining relations of SL_q(2) arising from the representations of U_q(sl₂(C)) by the R-matrix. The author gives the realization of O(Sp_q(2n)) arising from the representations of U_q(sp(2n)) via Jantzen approaching. For all u_ij, as the generators of the O(Sp_q(N)), we haveLetM,M,M' finite dimensional U_q(sp(2n))-modules,we haveThen, if we just consider C²ⁿ, as the module of natural representation of U_q(sp(2n)), we can get a Hopf algebra generated by all the matrix coefficients C_ij , and we have If we want to get all the relations of O(Sp_q(2n)) from c_ij, it is enough to show the R-matrices of (0.2) is the same as (0.4). We can obtain the R-matrices of (0.2) fromwhere However, the other R = (?) o f o P, hence, the key is the realization of (?) to get the R-matrix. Although Jantzen in [Ja] gave a satisfying result, it is not work to our case. Then the author gives a general formular of (?)' to the case that we consider now:such that (?) o f o P - (?)' o f o P. Consider C²ⁿ, as the module of natural representation of U_q(sp{2n)), by means of choosing a good base,such that C²(n-1), as the module of natural representation of U_q(sp(2(n-1))), is a submodule of C²ⁿ, then, we can show the result by induction on n.